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I'm a psycholinguistics student with few knowledge in statistics and I have some doubts about a Correlation of Fixed Effects in lmer function (lme4 package).

My response variable is RT (Reaction Time in a self-paced reading experiment) and my independent variables are Ant (PP, NP) and Verbo (SG, PL).

I have modeled the data with intercepts for Sujeitos (the people who are doing the task) and Item (the sentences i've used), asking for principal effects and interactions between the variables. Here is the model lmer(RT~Ant*Verbo+(1|Sujeitos)+(1|Item)) and that's the coefficients for fixed effects:

Table Fixed Effects

So, I have made a table of the coefficients for the interactions.

My problem is: that -0.705 and -0.716 correlation effects are a problem for my interaction terms? I'm saying this because the coefficients for the condition NP:SG came from the coefficients of SG only:

NP:SG

and the coefficients for PP:PL came from the PP only:

PP:PL

So, to me, there is no problem here, because I'm not contrasting PP:SG x PP (correlation = -0.705) and I'm not contrasting PP:SG x SG ((correlation = -0.716)). But I do that when getting the coefficients for PP:SG:

PP:SG

In the last case, there is a contrast between PP:SG x PP and SG. So, the correlation could be a problem. Is this correct? And, if so, how can I deal with this? I've read some thinks in Jaeger's blog and in this book: Howell, 2010. Statistical Methods for Psychology, but it doesn't help much.

Thank you.

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No, it is not a problem. The correlation arises from the fact that your two categorical predictors are entered as dummy codes, and are therefore not orthogonal to the interaction term, even in the case of balanced data. So we fully expect both the predictors themselves (i.e., the vectors of predictor values) and the parameter estimates to be correlated. It is exactly the same issue in fixed-effects-only regression using, for example, lm().

If you want, you can recode your categorical predictors using sum-to-zero contrasts (e.g., contr.helmert(2)), and the correlations between the interaction and simple effects should be lessened. But it shouldn't make any difference as far as the test of the interaction term is concerned.

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  • $\begingroup$ But, @Jake Westfall, it's that way even if I'm concerned the p-values. I'm saying that because the correlation could inflate de error term. So, the p-values could be anti-conservative. $\endgroup$ – Igor Costa Jun 6 '13 at 14:10
  • $\begingroup$ I understand. So refit the model under the contrast codes I suggested and see if you get a different answer. $\endgroup$ – Jake Westfall Jun 6 '13 at 17:11
  • $\begingroup$ I do that, using contr.helmert(2) for both categorical variables and the output was completely different: the correlation was reduced to almost zero values, but the coefficients changed to. The standard error was reduced to. But I'll analyse this data another hour. Thank you very much. $\endgroup$ – Igor Costa Jun 6 '13 at 21:03
  • $\begingroup$ The simple effects will be different, but the interaction should be the same. Is this what you see? $\endgroup$ – Jake Westfall Jun 6 '13 at 23:26
  • $\begingroup$ Yes, it is! But I'm reading somethings in Crawley's R book about Helmert contrasts and trying to understand what I'm doing here. I'm saying that because the effects of AntPP became very significative. It could be that way because Helmert contrasts the first level NP against all other levels. But in that case I have only two levels (NP and PP), so "all other" are only PP and the result should be the same. But... I didn't undesrtand it very well (yet). $\endgroup$ – Igor Costa Jun 8 '13 at 11:45

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